3.177 \(\int \frac{(c i+d i x)^2 (A+B \log (e (\frac{a+b x}{c+d x})^n))^2}{(a g+b g x)^6} \, dx\)

Optimal. Leaf size=493 \[ -\frac{b^2 i^2 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 g^6 (a+b x)^5 (b c-a d)^3}-\frac{2 b^2 B i^2 n (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{25 g^6 (a+b x)^5 (b c-a d)^3}-\frac{d^2 i^2 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g^6 (a+b x)^3 (b c-a d)^3}-\frac{2 B d^2 i^2 n (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^6 (a+b x)^3 (b c-a d)^3}+\frac{b d i^2 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^6 (a+b x)^4 (b c-a d)^3}+\frac{b B d i^2 n (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 g^6 (a+b x)^4 (b c-a d)^3}-\frac{2 b^2 B^2 i^2 n^2 (c+d x)^5}{125 g^6 (a+b x)^5 (b c-a d)^3}-\frac{2 B^2 d^2 i^2 n^2 (c+d x)^3}{27 g^6 (a+b x)^3 (b c-a d)^3}+\frac{b B^2 d i^2 n^2 (c+d x)^4}{16 g^6 (a+b x)^4 (b c-a d)^3} \]

[Out]

(-2*B^2*d^2*i^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B^2*d*i^2*n^2*(c + d*x)^4)/(16*(b*c -
a*d)^3*g^6*(a + b*x)^4) - (2*b^2*B^2*i^2*n^2*(c + d*x)^5)/(125*(b*c - a*d)^3*g^6*(a + b*x)^5) - (2*B*d^2*i^2*
n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B*d*i^2*n*(c + d*
x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(b*c - a*d)^3*g^6*(a + b*x)^4) - (2*b^2*B*i^2*n*(c + d*x)^5*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]))/(25*(b*c - a*d)^3*g^6*(a + b*x)^5) - (d^2*i^2*(c + d*x)^3*(A + B*Log[e*(
(a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*d*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(
c + d*x))^n])^2)/(2*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*i^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n
])^2)/(5*(b*c - a*d)^3*g^6*(a + b*x)^5)

________________________________________________________________________________________

Rubi [C]  time = 4.31635, antiderivative size = 1085, normalized size of antiderivative = 2.2, number of steps used = 110, number of rules used = 11, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.244, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B^2 i^2 n^2 \log ^2(a+b x) d^5}{30 b^3 (b c-a d)^3 g^6}+\frac{B^2 i^2 n^2 \log ^2(c+d x) d^5}{30 b^3 (b c-a d)^3 g^6}-\frac{47 B^2 i^2 n^2 \log (a+b x) d^5}{900 b^3 (b c-a d)^3 g^6}-\frac{B i^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^5}{15 b^3 (b c-a d)^3 g^6}+\frac{47 B^2 i^2 n^2 \log (c+d x) d^5}{900 b^3 (b c-a d)^3 g^6}-\frac{B^2 i^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^5}{15 b^3 (b c-a d)^3 g^6}+\frac{B i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) d^5}{15 b^3 (b c-a d)^3 g^6}-\frac{B^2 i^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^5}{15 b^3 (b c-a d)^3 g^6}-\frac{B^2 i^2 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^5}{15 b^3 (b c-a d)^3 g^6}-\frac{B^2 i^2 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^5}{15 b^3 (b c-a d)^3 g^6}-\frac{B i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^4}{15 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{47 B^2 i^2 n^2 d^4}{900 b^3 (b c-a d)^2 g^6 (a+b x)}+\frac{B i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^3}{30 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{13 B^2 i^2 n^2 d^3}{1800 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{i^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 d^2}{3 b^3 g^6 (a+b x)^3}-\frac{B i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^2}{45 b^3 g^6 (a+b x)^3}+\frac{43 B^2 i^2 n^2 d^2}{2700 b^3 g^6 (a+b x)^3}-\frac{(b c-a d) i^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 d}{2 b^3 g^6 (a+b x)^4}-\frac{3 B (b c-a d) i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d}{20 b^3 g^6 (a+b x)^4}-\frac{7 B^2 (b c-a d) i^2 n^2 d}{400 b^3 g^6 (a+b x)^4}-\frac{(b c-a d)^2 i^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{2 B (b c-a d)^2 i^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac{2 B^2 (b c-a d)^2 i^2 n^2}{125 b^3 g^6 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g + b*g*x)^6,x]

[Out]

(-2*B^2*(b*c - a*d)^2*i^2*n^2)/(125*b^3*g^6*(a + b*x)^5) - (7*B^2*d*(b*c - a*d)*i^2*n^2)/(400*b^3*g^6*(a + b*x
)^4) + (43*B^2*d^2*i^2*n^2)/(2700*b^3*g^6*(a + b*x)^3) - (13*B^2*d^3*i^2*n^2)/(1800*b^3*(b*c - a*d)*g^6*(a + b
*x)^2) - (47*B^2*d^4*i^2*n^2)/(900*b^3*(b*c - a*d)^2*g^6*(a + b*x)) - (47*B^2*d^5*i^2*n^2*Log[a + b*x])/(900*b
^3*(b*c - a*d)^3*g^6) + (B^2*d^5*i^2*n^2*Log[a + b*x]^2)/(30*b^3*(b*c - a*d)^3*g^6) - (2*B*(b*c - a*d)^2*i^2*n
*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(25*b^3*g^6*(a + b*x)^5) - (3*B*d*(b*c - a*d)*i^2*n*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]))/(20*b^3*g^6*(a + b*x)^4) - (B*d^2*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(45*b^3
*g^6*(a + b*x)^3) + (B*d^3*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b^3*(b*c - a*d)*g^6*(a + b*x)^2)
- (B*d^4*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*b^3*(b*c - a*d)^2*g^6*(a + b*x)) - (B*d^5*i^2*n*Log
[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*b^3*(b*c - a*d)^3*g^6) - ((b*c - a*d)^2*i^2*(A + B*Log[e
*((a + b*x)/(c + d*x))^n])^2)/(5*b^3*g^6*(a + b*x)^5) - (d*(b*c - a*d)*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^
n])^2)/(2*b^3*g^6*(a + b*x)^4) - (d^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b^3*g^6*(a + b*x)^3) +
(47*B^2*d^5*i^2*n^2*Log[c + d*x])/(900*b^3*(b*c - a*d)^3*g^6) - (B^2*d^5*i^2*n^2*Log[-((d*(a + b*x))/(b*c - a*
d))]*Log[c + d*x])/(15*b^3*(b*c - a*d)^3*g^6) + (B*d^5*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*
x])/(15*b^3*(b*c - a*d)^3*g^6) + (B^2*d^5*i^2*n^2*Log[c + d*x]^2)/(30*b^3*(b*c - a*d)^3*g^6) - (B^2*d^5*i^2*n^
2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(15*b^3*(b*c - a*d)^3*g^6) - (B^2*d^5*i^2*n^2*PolyLog[2, -((d*(
a + b*x))/(b*c - a*d))])/(15*b^3*(b*c - a*d)^3*g^6) - (B^2*d^5*i^2*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/
(15*b^3*(b*c - a*d)^3*g^6)

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{(177 c+177 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx &=\int \left (\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^6 (a+b x)^6}+\frac{62658 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^6 (a+b x)^5}+\frac{31329 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^6 (a+b x)^4}\right ) \, dx\\ &=\frac{\left (31329 d^2\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4} \, dx}{b^2 g^6}+\frac{(62658 d (b c-a d)) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^5} \, dx}{b^2 g^6}+\frac{\left (31329 (b c-a d)^2\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^6} \, dx}{b^2 g^6}\\ &=-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{\left (20886 B d^2 n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac{(31329 B d (b c-a d) n) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^6}+\frac{\left (62658 B (b c-a d)^2 n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{\left (20886 B d^2 (b c-a d) n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac{\left (31329 B d (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^6}+\frac{\left (62658 B (b c-a d)^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{\left (20886 B d^2 (b c-a d) n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^4}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^6}+\frac{\left (31329 B d (b c-a d)^2 n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^5}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^6}+\frac{\left (62658 B (b c-a d)^3 n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^6}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^5}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^4}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)^3}+\frac{b d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (a+b x)^2}-\frac{b d^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^6 (a+b x)}+\frac{d^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^6 (c+d x)}\right ) \, dx}{5 b^3 g^6}\\ &=-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{\left (62658 B d^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{5 b^2 g^6}+\frac{\left (20886 B d^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^2 g^6}-\frac{\left (31329 B d^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^2 g^6}-\frac{\left (62658 B d^5 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}-\frac{\left (20886 B d^5 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}+\frac{\left (31329 B d^5 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}+\frac{\left (62658 B d^6 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B d^6 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B d^6 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B d^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{5 b^2 (b c-a d)^2 g^6}+\frac{\left (20886 B d^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d)^2 g^6}-\frac{\left (31329 B d^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d)^2 g^6}-\frac{\left (62658 B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{5 b^2 (b c-a d) g^6}-\frac{\left (20886 B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 (b c-a d) g^6}+\frac{\left (31329 B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 (b c-a d) g^6}-\frac{(62658 B d (b c-a d) n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^5} \, dx}{5 b^2 g^6}+\frac{(31329 B d (b c-a d) n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^5} \, dx}{b^2 g^6}+\frac{\left (62658 B (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^6} \, dx}{5 b^2 g^6}\\ &=-\frac{62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac{93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac{3481 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac{10443 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{10443 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{10443 B d^5 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{5 b^3 g^6}+\frac{\left (6962 B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}-\frac{\left (10443 B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac{\left (62658 B^2 d^5 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{5 b^3 (b c-a d)^3 g^6}-\frac{\left (62658 B^2 d^5 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^5 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d)^3 g^6}-\frac{\left (20886 B^2 d^5 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^5 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac{\left (31329 B^2 d^5 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^4 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{5 b^3 (b c-a d)^2 g^6}+\frac{\left (20886 B^2 d^4 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d)^2 g^6}-\frac{\left (31329 B^2 d^4 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d)^2 g^6}-\frac{\left (31329 B^2 d^3 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{5 b^3 (b c-a d) g^6}-\frac{\left (10443 B^2 d^3 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 (b c-a d) g^6}+\frac{\left (31329 B^2 d^3 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 (b c-a d) g^6}-\frac{\left (31329 B^2 d (b c-a d) n^2\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{10 b^3 g^6}+\frac{\left (31329 B^2 d (b c-a d) n^2\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^6}+\frac{\left (62658 B^2 (b c-a d)^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^6 (c+d x)} \, dx}{25 b^3 g^6}\\ &=-\frac{62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac{93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac{3481 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac{10443 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{10443 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{10443 B d^5 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^3 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{5 b^3 g^6}-\frac{\left (10443 B^2 d^3 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^6}+\frac{\left (31329 B^2 d^3 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^6}+\frac{\left (62658 B^2 d^5 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{5 b^3 (b c-a d)^3 g^6}-\frac{\left (62658 B^2 d^5 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^5 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^3 g^6}-\frac{\left (20886 B^2 d^5 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^5 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^3 g^6}+\frac{\left (31329 B^2 d^5 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^4 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{5 b^3 (b c-a d) g^6}+\frac{\left (20886 B^2 d^4 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^6}-\frac{\left (31329 B^2 d^4 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^6}+\frac{\left (20886 B^2 d^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{5 b^3 g^6}+\frac{\left (6962 B^2 d^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}-\frac{\left (10443 B^2 d^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}-\frac{\left (31329 B^2 d (b c-a d)^2 n^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{10 b^3 g^6}+\frac{\left (31329 B^2 d (b c-a d)^2 n^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^6}+\frac{\left (62658 B^2 (b c-a d)^3 n^2\right ) \int \frac{1}{(a+b x)^6 (c+d x)} \, dx}{25 b^3 g^6}\\ &=-\frac{62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac{93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac{3481 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac{10443 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{10443 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{10443 B d^5 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^3 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{5 b^3 g^6}-\frac{\left (10443 B^2 d^3 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^6}+\frac{\left (31329 B^2 d^3 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^3 g^6}+\frac{\left (62658 B^2 d^5 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}-\frac{\left (62658 B^2 d^5 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^5 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}-\frac{\left (20886 B^2 d^5 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^5 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}+\frac{\left (31329 B^2 d^5 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}-\frac{\left (62658 B^2 d^6 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^6 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}-\frac{\left (20886 B^2 d^6 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^6 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac{\left (31329 B^2 d^6 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^6 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^4 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{5 b^3 (b c-a d) g^6}+\frac{\left (20886 B^2 d^4 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 (b c-a d) g^6}-\frac{\left (31329 B^2 d^4 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 (b c-a d) g^6}+\frac{\left (20886 B^2 d^2 (b c-a d) n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{5 b^3 g^6}+\frac{\left (6962 B^2 d^2 (b c-a d) n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^6}-\frac{\left (10443 B^2 d^2 (b c-a d) n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^6}-\frac{\left (31329 B^2 d (b c-a d)^2 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{10 b^3 g^6}+\frac{\left (31329 B^2 d (b c-a d)^2 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b^3 g^6}+\frac{\left (62658 B^2 (b c-a d)^3 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^6}-\frac{b d}{(b c-a d)^2 (a+b x)^5}+\frac{b d^2}{(b c-a d)^3 (a+b x)^4}-\frac{b d^3}{(b c-a d)^4 (a+b x)^3}+\frac{b d^4}{(b c-a d)^5 (a+b x)^2}-\frac{b d^5}{(b c-a d)^6 (a+b x)}+\frac{d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{25 b^3 g^6}\\ &=-\frac{62658 B^2 (b c-a d)^2 n^2}{125 b^3 g^6 (a+b x)^5}-\frac{219303 B^2 d (b c-a d) n^2}{400 b^3 g^6 (a+b x)^4}+\frac{149683 B^2 d^2 n^2}{300 b^3 g^6 (a+b x)^3}-\frac{45253 B^2 d^3 n^2}{200 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{163607 B^2 d^4 n^2}{100 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{163607 B^2 d^5 n^2 \log (a+b x)}{100 b^3 (b c-a d)^3 g^6}-\frac{62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac{93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac{3481 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac{10443 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{10443 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{163607 B^2 d^5 n^2 \log (c+d x)}{100 b^3 (b c-a d)^3 g^6}-\frac{10443 B^2 d^5 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac{10443 B d^5 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}-\frac{10443 B^2 d^5 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^5 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^5 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^5 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^6 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^6 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^6 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}\\ &=-\frac{62658 B^2 (b c-a d)^2 n^2}{125 b^3 g^6 (a+b x)^5}-\frac{219303 B^2 d (b c-a d) n^2}{400 b^3 g^6 (a+b x)^4}+\frac{149683 B^2 d^2 n^2}{300 b^3 g^6 (a+b x)^3}-\frac{45253 B^2 d^3 n^2}{200 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{163607 B^2 d^4 n^2}{100 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{163607 B^2 d^5 n^2 \log (a+b x)}{100 b^3 (b c-a d)^3 g^6}+\frac{10443 B^2 d^5 n^2 \log ^2(a+b x)}{10 b^3 (b c-a d)^3 g^6}-\frac{62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac{93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac{3481 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac{10443 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{10443 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{163607 B^2 d^5 n^2 \log (c+d x)}{100 b^3 (b c-a d)^3 g^6}-\frac{10443 B^2 d^5 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac{10443 B d^5 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac{10443 B^2 d^5 n^2 \log ^2(c+d x)}{10 b^3 (b c-a d)^3 g^6}-\frac{10443 B^2 d^5 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (62658 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^3 g^6}+\frac{\left (20886 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^3 g^6}-\frac{\left (31329 B^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^3 g^6}\\ &=-\frac{62658 B^2 (b c-a d)^2 n^2}{125 b^3 g^6 (a+b x)^5}-\frac{219303 B^2 d (b c-a d) n^2}{400 b^3 g^6 (a+b x)^4}+\frac{149683 B^2 d^2 n^2}{300 b^3 g^6 (a+b x)^3}-\frac{45253 B^2 d^3 n^2}{200 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{163607 B^2 d^4 n^2}{100 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{163607 B^2 d^5 n^2 \log (a+b x)}{100 b^3 (b c-a d)^3 g^6}+\frac{10443 B^2 d^5 n^2 \log ^2(a+b x)}{10 b^3 (b c-a d)^3 g^6}-\frac{62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac{93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac{3481 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac{10443 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac{10443 B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac{10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac{31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac{31329 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac{10443 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac{163607 B^2 d^5 n^2 \log (c+d x)}{100 b^3 (b c-a d)^3 g^6}-\frac{10443 B^2 d^5 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac{10443 B d^5 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac{10443 B^2 d^5 n^2 \log ^2(c+d x)}{10 b^3 (b c-a d)^3 g^6}-\frac{10443 B^2 d^5 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}-\frac{10443 B^2 d^5 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}-\frac{10443 B^2 d^5 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}\\ \end{align*}

Mathematica [C]  time = 3.96675, size = 2320, normalized size = 4.71 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g + b*g*x)^6,x]

[Out]

-(i^2*(10800*(b*c - a*d)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 27000*d*(b*c - a*d)^4*(a + b*x)*(A + B*L
og[e*((a + b*x)/(c + d*x))^n])^2 - 18000*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n
])^2 + 1000*B*d^2*n*(a + b*x)^2*(12*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 18*d*(b*c - a*d)^2*
(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 36*d^2*(b*c - a*d)*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]) + 36*d^3*(a + b*x)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 36*d^3*(a + b*x)^3*(A +
B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 36*B*d^2*n*(a + b*x)^2*(b*c - a*d + d*(a + b*x)*Log[a + b*x]
- d*(a + b*x)*Log[c + d*x]) - 9*B*d*n*(a + b*x)*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x
)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*n*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d)^2*(a + b*x) + 6*
d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]) - 18*B*d^3*n*(a
+ b*x)^3*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c)
+ a*d)]) + 18*B*d^3*n*(a + b*x)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*Poly
Log[2, (b*(c + d*x))/(b*c - a*d)])) + 375*B*d*n*(a + b*x)*(36*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))
^n]) + 48*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 72*d^2*(b*c - a*d)^2*(a + b*x)
^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*
x))^n]) - 144*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^4*(a + b*x)^4*(A + B
*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 144*B*d^3*n*(a + b*x)^3*(b*c - a*d + d*(a + b*x)*Log[a + b*x]
- d*(a + b*x)*Log[c + d*x]) + 36*B*d^2*n*(a + b*x)^2*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a
+ b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) - 8*B*d*n*(a + b*x)*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d)
^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]
) + 3*B*n*(3*(b*c - a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 12*d^3*(-(b*c)
+ a*d)*(a + b*x)^3 - 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x]) + 72*B*d^4*n*(a + b*x
)^4*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d
)]) - 72*B*d^4*n*(a + b*x)^4*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2,
(b*(c + d*x))/(b*c - a*d)])) + 6*B*n*(-225*a*B*d*(b*c - a*d)^4*n + 144*B*(b*c - a*d)^5*n - 225*b*B*d*(b*c - a
*d)^4*n*x + 300*a*B*d^2*(b*c - a*d)^3*n*(a + b*x) - 180*B*d*(b*c - a*d)^4*n*(a + b*x) + 300*b*B*d^2*(b*c - a*d
)^3*n*x*(a + b*x) - 450*a*B*d^3*(b*c - a*d)^2*n*(a + b*x)^2 + 640*B*d^2*(b*c - a*d)^3*n*(a + b*x)^2 - 450*b*B*
d^3*(b*c - a*d)^2*n*x*(a + b*x)^2 + 900*a*B*d^4*(b*c - a*d)*n*(a + b*x)^3 - 1860*B*d^3*(b*c - a*d)^2*n*(a + b*
x)^3 + 900*b*B*d^4*(b*c - a*d)*n*x*(a + b*x)^3 + 3600*b*B*c*d^4*n*(a + b*x)^4 - 3600*a*B*d^5*n*(a + b*x)^4 + 3
720*B*d^4*(b*c - a*d)*n*(a + b*x)^4 + 900*a*B*d^5*n*(a + b*x)^4*Log[a + b*x] + 900*b*B*d^5*n*x*(a + b*x)^4*Log
[a + b*x] + 7320*B*d^5*n*(a + b*x)^5*Log[a + b*x] + 720*(b*c - a*d)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) -
900*d*(b*c - a*d)^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 1200*d^2*(b*c - a*d)^3*(a + b*x)^2*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]) - 1800*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]
) + 3600*d^4*(b*c - a*d)*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 3600*d^5*(a + b*x)^5*Log[a + b*x
]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 900*a*B*d^5*n*(a + b*x)^4*Log[c + d*x] - 900*b*B*d^5*n*x*(a + b*x)^
4*Log[c + d*x] - 7320*B*d^5*n*(a + b*x)^5*Log[c + d*x] - 3600*d^5*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x
))^n])*Log[c + d*x] - 1800*B*d^5*n*(a + b*x)^5*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)])
- 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 1800*B*d^5*n*(a + b*x)^5*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)
] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))))/(54000*b^3*(b*c - a*d)^3*g^6*(a +
b*x)^5)

________________________________________________________________________________________

Maple [F]  time = 0.733, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{ \left ( bgx+ag \right ) ^{6}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x)

[Out]

int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x)

________________________________________________________________________________________

Maxima [B]  time = 6.13958, size = 14764, normalized size = 29.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x, algorithm="maxima")

[Out]

-1/150*A*B*c^2*i^2*n*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*a^2*b^2*c^2*d^2 - 163*a^3*b*c*d^3 +
137*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*
(3*b^4*c^3*d - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c
^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b
^6*c*d^3 + a^5*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^
6*b^4*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^
6*x^2 + 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d^4)*g^6*x + (a^5*b^5
*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5
- 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6) - 60*d^5*log(d*
x + c)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6)
) - 1/900*A*B*d^2*i^2*n*((47*a^2*b^4*c^4 - 278*a^3*b^3*c^3*d + 822*a^4*b^2*c^2*d^2 - 278*a^5*b*c*d^3 + 47*a^6*
d^4 + 60*(10*b^6*c^2*d^2 - 5*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 - 30*(10*b^6*c^3*d - 95*a*b^5*c^2*d^2 + 46*a^2*b^4
*c*d^3 - 9*a^3*b^3*d^4)*x^3 + 10*(20*b^6*c^4 - 140*a*b^5*c^3*d + 537*a^2*b^4*c^2*d^2 - 248*a^3*b^3*c*d^3 + 47*
a^4*b^2*d^4)*x^2 + 5*(35*a*b^5*c^4 - 218*a^2*b^4*c^3*d + 702*a^3*b^3*c^2*d^2 - 278*a^4*b^2*c*d^3 + 47*a^5*b*d^
4)*x)/((b^12*c^4 - 4*a*b^11*c^3*d + 6*a^2*b^10*c^2*d^2 - 4*a^3*b^9*c*d^3 + a^4*b^8*d^4)*g^6*x^5 + 5*(a*b^11*c^
4 - 4*a^2*b^10*c^3*d + 6*a^3*b^9*c^2*d^2 - 4*a^4*b^8*c*d^3 + a^5*b^7*d^4)*g^6*x^4 + 10*(a^2*b^10*c^4 - 4*a^3*b
^9*c^3*d + 6*a^4*b^8*c^2*d^2 - 4*a^5*b^7*c*d^3 + a^6*b^6*d^4)*g^6*x^3 + 10*(a^3*b^9*c^4 - 4*a^4*b^8*c^3*d + 6*
a^5*b^7*c^2*d^2 - 4*a^6*b^6*c*d^3 + a^7*b^5*d^4)*g^6*x^2 + 5*(a^4*b^8*c^4 - 4*a^5*b^7*c^3*d + 6*a^6*b^6*c^2*d^
2 - 4*a^7*b^5*c*d^3 + a^8*b^4*d^4)*g^6*x + (a^5*b^7*c^4 - 4*a^6*b^6*c^3*d + 6*a^7*b^5*c^2*d^2 - 4*a^8*b^4*c*d^
3 + a^9*b^3*d^4)*g^6) + 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(b*x + a)/((b^8*c^5 - 5*a*b^7*c^4*d + 1
0*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6) - 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^
4 + a^2*d^5)*log(d*x + c)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^
4 - a^5*b^3*d^5)*g^6)) - 1/300*A*B*c*d*i^2*n*((27*a*b^4*c^4 - 148*a^2*b^3*c^3*d + 352*a^3*b^2*c^2*d^2 - 548*a^
4*b*c*d^3 + 77*a^5*d^4 - 60*(5*b^5*c*d^3 - a*b^4*d^4)*x^4 + 30*(5*b^5*c^2*d^2 - 46*a*b^4*c*d^3 + 9*a^2*b^3*d^4
)*x^3 - 10*(10*b^5*c^3*d - 67*a*b^4*c^2*d^2 + 248*a^2*b^3*c*d^3 - 47*a^3*b^2*d^4)*x^2 + 5*(15*b^5*c^4 - 88*a*b
^4*c^3*d + 232*a^2*b^3*c^2*d^2 - 428*a^3*b^2*c*d^3 + 77*a^4*b*d^4)*x)/((b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*
c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4)*g^6*x^5 + 5*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4
*b^7*c*d^3 + a^5*b^6*d^4)*g^6*x^4 + 10*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 +
a^6*b^5*d^4)*g^6*x^3 + 10*(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4)*
g^6*x^2 + 5*(a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*g^6*x + (a^5*b
^6*c^4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4*c^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*g^6) - 60*(5*b*c*d^4 - a*d^5)*lo
g(b*x + a)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5
)*g^6) + 60*(5*b*c*d^4 - a*d^5)*log(d*x + c)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d
^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6)) - 1/10*(5*b*x + a)*B^2*c*d*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n
)^2/(b^7*g^6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6)
- 1/30*(10*b^2*x^2 + 5*a*b*x + a^2)*B^2*d^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2/(b^8*g^6*x^5 + 5*a*b^
7*g^6*x^4 + 10*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) - 1/9000*(60*n*((60*b^4*d
^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*a^2*b^2*c^2*d^2 - 163*a^3*b*c*d^3 + 137*a^4*d^4 - 30*(b^4*c*d^3 - 9
*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*(3*b^4*c^3*d - 17*a*b^3*c^2*d^2
+ 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*
b^6*d^4)*g^6*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*g^6*x^4
+ 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*g^6*x^3 + 10*(a^3*b^
7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^6*x^2 + 5*(a^4*b^6*c^4 - 4*a^5*
b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d^4)*g^6*x + (a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*
b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c
^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6) - 60*d^5*log(d*x + c)/((b^6*c^5 - 5*a*b^5*c^4*
d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6))*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n) + (144*b^5*c^5 - 1125*a*b^4*c^4*d + 4000*a^2*b^3*c^3*d^2 - 9000*a^3*b^2*c^2*d^3 + 18000*a^4*b*c*d^4
- 12019*a^5*d^5 + 8220*(b^5*c*d^4 - a*b^4*d^5)*x^4 - 30*(77*b^5*c^2*d^3 - 1250*a*b^4*c*d^4 + 1173*a^2*b^3*d^5)
*x^3 + 10*(94*b^5*c^3*d^2 - 975*a*b^4*c^2*d^3 + 6600*a^2*b^3*c*d^4 - 5719*a^3*b^2*d^5)*x^2 - 1800*(b^5*d^5*x^5
+ 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^5)*log(b*x + a)^2 - 1800*
(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^5)*log(d*x +
c)^2 - 5*(81*b^5*c^4*d - 700*a*b^4*c^3*d^2 + 3000*a^2*b^3*c^2*d^3 - 10800*a^3*b^2*c*d^4 + 8419*a^4*b*d^5)*x +
8220*(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^5)*log(b
*x + a) - 60*(137*b^5*d^5*x^5 + 685*a*b^4*d^5*x^4 + 1370*a^2*b^3*d^5*x^3 + 1370*a^3*b^2*d^5*x^2 + 685*a^4*b*d^
5*x + 137*a^5*d^5 - 60*(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*
x + a^5*d^5)*log(b*x + a))*log(d*x + c))*n^2/(a^5*b^6*c^5*g^6 - 5*a^6*b^5*c^4*d*g^6 + 10*a^7*b^4*c^3*d^2*g^6 -
10*a^8*b^3*c^2*d^3*g^6 + 5*a^9*b^2*c*d^4*g^6 - a^10*b*d^5*g^6 + (b^11*c^5*g^6 - 5*a*b^10*c^4*d*g^6 + 10*a^2*b
^9*c^3*d^2*g^6 - 10*a^3*b^8*c^2*d^3*g^6 + 5*a^4*b^7*c*d^4*g^6 - a^5*b^6*d^5*g^6)*x^5 + 5*(a*b^10*c^5*g^6 - 5*a
^2*b^9*c^4*d*g^6 + 10*a^3*b^8*c^3*d^2*g^6 - 10*a^4*b^7*c^2*d^3*g^6 + 5*a^5*b^6*c*d^4*g^6 - a^6*b^5*d^5*g^6)*x^
4 + 10*(a^2*b^9*c^5*g^6 - 5*a^3*b^8*c^4*d*g^6 + 10*a^4*b^7*c^3*d^2*g^6 - 10*a^5*b^6*c^2*d^3*g^6 + 5*a^6*b^5*c*
d^4*g^6 - a^7*b^4*d^5*g^6)*x^3 + 10*(a^3*b^8*c^5*g^6 - 5*a^4*b^7*c^4*d*g^6 + 10*a^5*b^6*c^3*d^2*g^6 - 10*a^6*b
^5*c^2*d^3*g^6 + 5*a^7*b^4*c*d^4*g^6 - a^8*b^3*d^5*g^6)*x^2 + 5*(a^4*b^7*c^5*g^6 - 5*a^5*b^6*c^4*d*g^6 + 10*a^
6*b^5*c^3*d^2*g^6 - 10*a^7*b^4*c^2*d^3*g^6 + 5*a^8*b^3*c*d^4*g^6 - a^9*b^2*d^5*g^6)*x))*B^2*c^2*i^2 - 1/18000*
(60*n*((27*a*b^4*c^4 - 148*a^2*b^3*c^3*d + 352*a^3*b^2*c^2*d^2 - 548*a^4*b*c*d^3 + 77*a^5*d^4 - 60*(5*b^5*c*d^
3 - a*b^4*d^4)*x^4 + 30*(5*b^5*c^2*d^2 - 46*a*b^4*c*d^3 + 9*a^2*b^3*d^4)*x^3 - 10*(10*b^5*c^3*d - 67*a*b^4*c^2
*d^2 + 248*a^2*b^3*c*d^3 - 47*a^3*b^2*d^4)*x^2 + 5*(15*b^5*c^4 - 88*a*b^4*c^3*d + 232*a^2*b^3*c^2*d^2 - 428*a^
3*b^2*c*d^3 + 77*a^4*b*d^4)*x)/((b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4
)*g^6*x^5 + 5*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4*b^7*c*d^3 + a^5*b^6*d^4)*g^6*x^4 + 10*
(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6*b^5*d^4)*g^6*x^3 + 10*(a^3*b^8*c^4
- 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4)*g^6*x^2 + 5*(a^4*b^7*c^4 - 4*a^5*b^6*c^
3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*g^6*x + (a^5*b^6*c^4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4*c^
2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*g^6) - 60*(5*b*c*d^4 - a*d^5)*log(b*x + a)/((b^7*c^5 - 5*a*b^6*c^4*d +
10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6) + 60*(5*b*c*d^4 - a*d^5)*log(d*x
+ c)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6
))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + (549*a*b^5*c^5 - 4625*a^2*b^4*c^4*d + 19000*a^3*b^3*c^3*d^2 - 6300
0*a^4*b^2*c^2*d^3 + 51875*a^5*b*c*d^4 - 3799*a^6*d^5 - 60*(625*b^6*c^2*d^3 - 702*a*b^5*c*d^4 + 77*a^2*b^4*d^5)
*x^4 + 30*(325*b^6*c^3*d^2 - 5667*a*b^5*c^2*d^3 + 5975*a^2*b^4*c*d^4 - 633*a^3*b^3*d^5)*x^3 - 10*(350*b^6*c^4*
d - 3949*a*b^5*c^3*d^2 + 29475*a^2*b^4*c^2*d^3 - 28775*a^3*b^3*c*d^4 + 2899*a^4*b^2*d^5)*x^2 + 1800*(5*a^5*b*c
*d^4 - a^6*d^5 + (5*b^6*c*d^4 - a*b^5*d^5)*x^5 + 5*(5*a*b^5*c*d^4 - a^2*b^4*d^5)*x^4 + 10*(5*a^2*b^4*c*d^4 - a
^3*b^3*d^5)*x^3 + 10*(5*a^3*b^3*c*d^4 - a^4*b^2*d^5)*x^2 + 5*(5*a^4*b^2*c*d^4 - a^5*b*d^5)*x)*log(b*x + a)^2 +
1800*(5*a^5*b*c*d^4 - a^6*d^5 + (5*b^6*c*d^4 - a*b^5*d^5)*x^5 + 5*(5*a*b^5*c*d^4 - a^2*b^4*d^5)*x^4 + 10*(5*a
^2*b^4*c*d^4 - a^3*b^3*d^5)*x^3 + 10*(5*a^3*b^3*c*d^4 - a^4*b^2*d^5)*x^2 + 5*(5*a^4*b^2*c*d^4 - a^5*b*d^5)*x)*
log(d*x + c)^2 + 5*(225*b^6*c^5 - 2201*a*b^5*c^4*d + 10900*a^2*b^4*c^3*d^2 - 46200*a^3*b^3*c^2*d^3 + 41075*a^4
*b^2*c*d^4 - 3799*a^5*b*d^5)*x - 60*(625*a^5*b*c*d^4 - 77*a^6*d^5 + (625*b^6*c*d^4 - 77*a*b^5*d^5)*x^5 + 5*(62
5*a*b^5*c*d^4 - 77*a^2*b^4*d^5)*x^4 + 10*(625*a^2*b^4*c*d^4 - 77*a^3*b^3*d^5)*x^3 + 10*(625*a^3*b^3*c*d^4 - 77
*a^4*b^2*d^5)*x^2 + 5*(625*a^4*b^2*c*d^4 - 77*a^5*b*d^5)*x)*log(b*x + a) + 60*(625*a^5*b*c*d^4 - 77*a^6*d^5 +
(625*b^6*c*d^4 - 77*a*b^5*d^5)*x^5 + 5*(625*a*b^5*c*d^4 - 77*a^2*b^4*d^5)*x^4 + 10*(625*a^2*b^4*c*d^4 - 77*a^3
*b^3*d^5)*x^3 + 10*(625*a^3*b^3*c*d^4 - 77*a^4*b^2*d^5)*x^2 + 5*(625*a^4*b^2*c*d^4 - 77*a^5*b*d^5)*x - 60*(5*a
^5*b*c*d^4 - a^6*d^5 + (5*b^6*c*d^4 - a*b^5*d^5)*x^5 + 5*(5*a*b^5*c*d^4 - a^2*b^4*d^5)*x^4 + 10*(5*a^2*b^4*c*d
^4 - a^3*b^3*d^5)*x^3 + 10*(5*a^3*b^3*c*d^4 - a^4*b^2*d^5)*x^2 + 5*(5*a^4*b^2*c*d^4 - a^5*b*d^5)*x)*log(b*x +
a))*log(d*x + c))*n^2/(a^5*b^7*c^5*g^6 - 5*a^6*b^6*c^4*d*g^6 + 10*a^7*b^5*c^3*d^2*g^6 - 10*a^8*b^4*c^2*d^3*g^6
+ 5*a^9*b^3*c*d^4*g^6 - a^10*b^2*d^5*g^6 + (b^12*c^5*g^6 - 5*a*b^11*c^4*d*g^6 + 10*a^2*b^10*c^3*d^2*g^6 - 10*
a^3*b^9*c^2*d^3*g^6 + 5*a^4*b^8*c*d^4*g^6 - a^5*b^7*d^5*g^6)*x^5 + 5*(a*b^11*c^5*g^6 - 5*a^2*b^10*c^4*d*g^6 +
10*a^3*b^9*c^3*d^2*g^6 - 10*a^4*b^8*c^2*d^3*g^6 + 5*a^5*b^7*c*d^4*g^6 - a^6*b^6*d^5*g^6)*x^4 + 10*(a^2*b^10*c^
5*g^6 - 5*a^3*b^9*c^4*d*g^6 + 10*a^4*b^8*c^3*d^2*g^6 - 10*a^5*b^7*c^2*d^3*g^6 + 5*a^6*b^6*c*d^4*g^6 - a^7*b^5*
d^5*g^6)*x^3 + 10*(a^3*b^9*c^5*g^6 - 5*a^4*b^8*c^4*d*g^6 + 10*a^5*b^7*c^3*d^2*g^6 - 10*a^6*b^6*c^2*d^3*g^6 + 5
*a^7*b^5*c*d^4*g^6 - a^8*b^4*d^5*g^6)*x^2 + 5*(a^4*b^8*c^5*g^6 - 5*a^5*b^7*c^4*d*g^6 + 10*a^6*b^6*c^3*d^2*g^6
- 10*a^7*b^5*c^2*d^3*g^6 + 5*a^8*b^4*c*d^4*g^6 - a^9*b^3*d^5*g^6)*x))*B^2*c*d*i^2 - 1/54000*(60*n*((47*a^2*b^4
*c^4 - 278*a^3*b^3*c^3*d + 822*a^4*b^2*c^2*d^2 - 278*a^5*b*c*d^3 + 47*a^6*d^4 + 60*(10*b^6*c^2*d^2 - 5*a*b^5*c
*d^3 + a^2*b^4*d^4)*x^4 - 30*(10*b^6*c^3*d - 95*a*b^5*c^2*d^2 + 46*a^2*b^4*c*d^3 - 9*a^3*b^3*d^4)*x^3 + 10*(20
*b^6*c^4 - 140*a*b^5*c^3*d + 537*a^2*b^4*c^2*d^2 - 248*a^3*b^3*c*d^3 + 47*a^4*b^2*d^4)*x^2 + 5*(35*a*b^5*c^4 -
218*a^2*b^4*c^3*d + 702*a^3*b^3*c^2*d^2 - 278*a^4*b^2*c*d^3 + 47*a^5*b*d^4)*x)/((b^12*c^4 - 4*a*b^11*c^3*d +
6*a^2*b^10*c^2*d^2 - 4*a^3*b^9*c*d^3 + a^4*b^8*d^4)*g^6*x^5 + 5*(a*b^11*c^4 - 4*a^2*b^10*c^3*d + 6*a^3*b^9*c^2
*d^2 - 4*a^4*b^8*c*d^3 + a^5*b^7*d^4)*g^6*x^4 + 10*(a^2*b^10*c^4 - 4*a^3*b^9*c^3*d + 6*a^4*b^8*c^2*d^2 - 4*a^5
*b^7*c*d^3 + a^6*b^6*d^4)*g^6*x^3 + 10*(a^3*b^9*c^4 - 4*a^4*b^8*c^3*d + 6*a^5*b^7*c^2*d^2 - 4*a^6*b^6*c*d^3 +
a^7*b^5*d^4)*g^6*x^2 + 5*(a^4*b^8*c^4 - 4*a^5*b^7*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^7*b^5*c*d^3 + a^8*b^4*d^4)*g
^6*x + (a^5*b^7*c^4 - 4*a^6*b^6*c^3*d + 6*a^7*b^5*c^2*d^2 - 4*a^8*b^4*c*d^3 + a^9*b^3*d^4)*g^6) + 60*(10*b^2*c
^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(b*x + a)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d
^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6) - 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(d*x + c)/((b^8*c^5
- 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6))*log(e*(b*x/(d
*x + c) + a/(d*x + c))^n) + (1489*a^2*b^5*c^5 - 14375*a^3*b^4*c^4*d + 85000*a^4*b^3*c^3*d^2 - 85000*a^5*b^2*c^
2*d^3 + 14375*a^6*b*c*d^4 - 1489*a^7*d^5 + 60*(1100*b^7*c^3*d^2 - 1425*a*b^6*c^2*d^3 + 372*a^2*b^5*c*d^4 - 47*
a^3*b^4*d^5)*x^4 - 30*(500*b^7*c^4*d - 9825*a*b^6*c^3*d^2 + 11937*a^2*b^5*c^2*d^3 - 2975*a^3*b^4*c*d^4 + 363*a
^4*b^3*d^5)*x^3 + 10*(400*b^7*c^5 - 5450*a*b^6*c^4*d + 49189*a^2*b^5*c^3*d^2 - 55525*a^3*b^4*c^2*d^3 + 12875*a
^4*b^3*c*d^4 - 1489*a^5*b^2*d^5)*x^2 - 1800*(10*a^5*b^2*c^2*d^3 - 5*a^6*b*c*d^4 + a^7*d^5 + (10*b^7*c^2*d^3 -
5*a*b^6*c*d^4 + a^2*b^5*d^5)*x^5 + 5*(10*a*b^6*c^2*d^3 - 5*a^2*b^5*c*d^4 + a^3*b^4*d^5)*x^4 + 10*(10*a^2*b^5*c
^2*d^3 - 5*a^3*b^4*c*d^4 + a^4*b^3*d^5)*x^3 + 10*(10*a^3*b^4*c^2*d^3 - 5*a^4*b^3*c*d^4 + a^5*b^2*d^5)*x^2 + 5*
(10*a^4*b^3*c^2*d^3 - 5*a^5*b^2*c*d^4 + a^6*b*d^5)*x)*log(b*x + a)^2 - 1800*(10*a^5*b^2*c^2*d^3 - 5*a^6*b*c*d^
4 + a^7*d^5 + (10*b^7*c^2*d^3 - 5*a*b^6*c*d^4 + a^2*b^5*d^5)*x^5 + 5*(10*a*b^6*c^2*d^3 - 5*a^2*b^5*c*d^4 + a^3
*b^4*d^5)*x^4 + 10*(10*a^2*b^5*c^2*d^3 - 5*a^3*b^4*c*d^4 + a^4*b^3*d^5)*x^3 + 10*(10*a^3*b^4*c^2*d^3 - 5*a^4*b
^3*c*d^4 + a^5*b^2*d^5)*x^2 + 5*(10*a^4*b^3*c^2*d^3 - 5*a^5*b^2*c*d^4 + a^6*b*d^5)*x)*log(d*x + c)^2 + 5*(925*
a*b^6*c^5 - 9911*a^2*b^5*c^4*d + 67900*a^3*b^4*c^3*d^2 - 71800*a^4*b^3*c^2*d^3 + 14375*a^5*b^2*c*d^4 - 1489*a^
6*b*d^5)*x + 60*(1100*a^5*b^2*c^2*d^3 - 325*a^6*b*c*d^4 + 47*a^7*d^5 + (1100*b^7*c^2*d^3 - 325*a*b^6*c*d^4 + 4
7*a^2*b^5*d^5)*x^5 + 5*(1100*a*b^6*c^2*d^3 - 325*a^2*b^5*c*d^4 + 47*a^3*b^4*d^5)*x^4 + 10*(1100*a^2*b^5*c^2*d^
3 - 325*a^3*b^4*c*d^4 + 47*a^4*b^3*d^5)*x^3 + 10*(1100*a^3*b^4*c^2*d^3 - 325*a^4*b^3*c*d^4 + 47*a^5*b^2*d^5)*x
^2 + 5*(1100*a^4*b^3*c^2*d^3 - 325*a^5*b^2*c*d^4 + 47*a^6*b*d^5)*x)*log(b*x + a) - 60*(1100*a^5*b^2*c^2*d^3 -
325*a^6*b*c*d^4 + 47*a^7*d^5 + (1100*b^7*c^2*d^3 - 325*a*b^6*c*d^4 + 47*a^2*b^5*d^5)*x^5 + 5*(1100*a*b^6*c^2*d
^3 - 325*a^2*b^5*c*d^4 + 47*a^3*b^4*d^5)*x^4 + 10*(1100*a^2*b^5*c^2*d^3 - 325*a^3*b^4*c*d^4 + 47*a^4*b^3*d^5)*
x^3 + 10*(1100*a^3*b^4*c^2*d^3 - 325*a^4*b^3*c*d^4 + 47*a^5*b^2*d^5)*x^2 + 5*(1100*a^4*b^3*c^2*d^3 - 325*a^5*b
^2*c*d^4 + 47*a^6*b*d^5)*x - 60*(10*a^5*b^2*c^2*d^3 - 5*a^6*b*c*d^4 + a^7*d^5 + (10*b^7*c^2*d^3 - 5*a*b^6*c*d^
4 + a^2*b^5*d^5)*x^5 + 5*(10*a*b^6*c^2*d^3 - 5*a^2*b^5*c*d^4 + a^3*b^4*d^5)*x^4 + 10*(10*a^2*b^5*c^2*d^3 - 5*a
^3*b^4*c*d^4 + a^4*b^3*d^5)*x^3 + 10*(10*a^3*b^4*c^2*d^3 - 5*a^4*b^3*c*d^4 + a^5*b^2*d^5)*x^2 + 5*(10*a^4*b^3*
c^2*d^3 - 5*a^5*b^2*c*d^4 + a^6*b*d^5)*x)*log(b*x + a))*log(d*x + c))*n^2/(a^5*b^8*c^5*g^6 - 5*a^6*b^7*c^4*d*g
^6 + 10*a^7*b^6*c^3*d^2*g^6 - 10*a^8*b^5*c^2*d^3*g^6 + 5*a^9*b^4*c*d^4*g^6 - a^10*b^3*d^5*g^6 + (b^13*c^5*g^6
- 5*a*b^12*c^4*d*g^6 + 10*a^2*b^11*c^3*d^2*g^6 - 10*a^3*b^10*c^2*d^3*g^6 + 5*a^4*b^9*c*d^4*g^6 - a^5*b^8*d^5*g
^6)*x^5 + 5*(a*b^12*c^5*g^6 - 5*a^2*b^11*c^4*d*g^6 + 10*a^3*b^10*c^3*d^2*g^6 - 10*a^4*b^9*c^2*d^3*g^6 + 5*a^5*
b^8*c*d^4*g^6 - a^6*b^7*d^5*g^6)*x^4 + 10*(a^2*b^11*c^5*g^6 - 5*a^3*b^10*c^4*d*g^6 + 10*a^4*b^9*c^3*d^2*g^6 -
10*a^5*b^8*c^2*d^3*g^6 + 5*a^6*b^7*c*d^4*g^6 - a^7*b^6*d^5*g^6)*x^3 + 10*(a^3*b^10*c^5*g^6 - 5*a^4*b^9*c^4*d*g
^6 + 10*a^5*b^8*c^3*d^2*g^6 - 10*a^6*b^7*c^2*d^3*g^6 + 5*a^7*b^6*c*d^4*g^6 - a^8*b^5*d^5*g^6)*x^2 + 5*(a^4*b^9
*c^5*g^6 - 5*a^5*b^8*c^4*d*g^6 + 10*a^6*b^7*c^3*d^2*g^6 - 10*a^7*b^6*c^2*d^3*g^6 + 5*a^8*b^5*c*d^4*g^6 - a^9*b
^4*d^5*g^6)*x))*B^2*d^2*i^2 - 1/5*(5*b*x + a)*A*B*c*d*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^7*g^6*x^5
+ 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) - 1/15*(10*b^2*x^
2 + 5*a*b*x + a^2)*A*B*d^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 10*a^2*
b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) - 1/5*B^2*c^2*i^2*log(e*(b*x/(d*x + c) + a/(
d*x + c))^n)^2/(b^6*g^6*x^5 + 5*a*b^5*g^6*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^3*g^6*x^2 + 5*a^4*b^2*g^6*x + a^
5*b*g^6) - 1/10*(5*b*x + a)*A^2*c*d*i^2/(b^7*g^6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x
^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) - 1/30*(10*b^2*x^2 + 5*a*b*x + a^2)*A^2*d^2*i^2/(b^8*g^6*x^5 + 5*a*b^7*g^6
*x^4 + 10*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) - 2/5*A*B*c^2*i^2*log(e*(b*x/(
d*x + c) + a/(d*x + c))^n)/(b^6*g^6*x^5 + 5*a*b^5*g^6*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^3*g^6*x^2 + 5*a^4*b^
2*g^6*x + a^5*b*g^6) - 1/5*A^2*c^2*i^2/(b^6*g^6*x^5 + 5*a*b^5*g^6*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^3*g^6*x^
2 + 5*a^4*b^2*g^6*x + a^5*b*g^6)

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Fricas [B]  time = 0.835333, size = 5374, normalized size = 10.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x, algorithm="fricas")

[Out]

-1/54000*((864*B^2*b^5*c^5 - 3375*B^2*a*b^4*c^4*d + 4000*B^2*a^2*b^3*c^3*d^2 - 1489*B^2*a^5*d^5)*i^2*n^2 + 60*
(47*(B^2*b^5*c*d^4 - B^2*a*b^4*d^5)*i^2*n^2 + 60*(A*B*b^5*c*d^4 - A*B*a*b^4*d^5)*i^2*n)*x^4 + 60*(72*A*B*b^5*c
^5 - 225*A*B*a*b^4*c^4*d + 200*A*B*a^2*b^3*c^3*d^2 - 47*A*B*a^5*d^5)*i^2*n + 30*((13*B^2*b^5*c^2*d^3 + 350*B^2
*a*b^4*c*d^4 - 363*B^2*a^2*b^3*d^5)*i^2*n^2 - 60*(A*B*b^5*c^2*d^3 - 10*A*B*a*b^4*c*d^4 + 9*A*B*a^2*b^3*d^5)*i^
2*n)*x^3 + 1800*(6*A^2*b^5*c^5 - 15*A^2*a*b^4*c^4*d + 10*A^2*a^2*b^3*c^3*d^2 - A^2*a^5*d^5)*i^2 - 10*((86*B^2*
b^5*c^3*d^2 - 375*B^2*a*b^4*c^2*d^3 - 1200*B^2*a^2*b^3*c*d^4 + 1489*B^2*a^3*b^2*d^5)*i^2*n^2 - 60*(2*A*B*b^5*c
^3*d^2 - 15*A*B*a*b^4*c^2*d^3 + 60*A*B*a^2*b^3*c*d^4 - 47*A*B*a^3*b^2*d^5)*i^2*n - 1800*(A^2*b^5*c^3*d^2 - 3*A
^2*a*b^4*c^2*d^3 + 3*A^2*a^2*b^3*c*d^4 - A^2*a^3*b^2*d^5)*i^2)*x^2 + 1800*(10*(B^2*b^5*c^3*d^2 - 3*B^2*a*b^4*c
^2*d^3 + 3*B^2*a^2*b^3*c*d^4 - B^2*a^3*b^2*d^5)*i^2*x^2 + 5*(3*B^2*b^5*c^4*d - 8*B^2*a*b^4*c^3*d^2 + 6*B^2*a^2
*b^3*c^2*d^3 - B^2*a^4*b*d^5)*i^2*x + (6*B^2*b^5*c^5 - 15*B^2*a*b^4*c^4*d + 10*B^2*a^2*b^3*c^3*d^2 - B^2*a^5*d
^5)*i^2)*log(e)^2 + 1800*(B^2*b^5*d^5*i^2*n^2*x^5 + 5*B^2*a*b^4*d^5*i^2*n^2*x^4 + 10*B^2*a^2*b^3*d^5*i^2*n^2*x
^3 + 10*(B^2*b^5*c^3*d^2 - 3*B^2*a*b^4*c^2*d^3 + 3*B^2*a^2*b^3*c*d^4)*i^2*n^2*x^2 + 5*(3*B^2*b^5*c^4*d - 8*B^2
*a*b^4*c^3*d^2 + 6*B^2*a^2*b^3*c^2*d^3)*i^2*n^2*x + (6*B^2*b^5*c^5 - 15*B^2*a*b^4*c^4*d + 10*B^2*a^2*b^3*c^3*d
^2)*i^2*n^2)*log((b*x + a)/(d*x + c))^2 + 5*((189*B^2*b^5*c^4*d - 1100*B^2*a*b^4*c^3*d^2 + 2400*B^2*a^2*b^3*c^
2*d^3 - 1489*B^2*a^4*b*d^5)*i^2*n^2 + 60*(27*A*B*b^5*c^4*d - 100*A*B*a*b^4*c^3*d^2 + 120*A*B*a^2*b^3*c^2*d^3 -
47*A*B*a^4*b*d^5)*i^2*n + 1800*(3*A^2*b^5*c^4*d - 8*A^2*a*b^4*c^3*d^2 + 6*A^2*a^2*b^3*c^2*d^3 - A^2*a^4*b*d^5
)*i^2)*x + 60*(60*(B^2*b^5*c*d^4 - B^2*a*b^4*d^5)*i^2*n*x^4 - 30*(B^2*b^5*c^2*d^3 - 10*B^2*a*b^4*c*d^4 + 9*B^2
*a^2*b^3*d^5)*i^2*n*x^3 + (72*B^2*b^5*c^5 - 225*B^2*a*b^4*c^4*d + 200*B^2*a^2*b^3*c^3*d^2 - 47*B^2*a^5*d^5)*i^
2*n + 60*(6*A*B*b^5*c^5 - 15*A*B*a*b^4*c^4*d + 10*A*B*a^2*b^3*c^3*d^2 - A*B*a^5*d^5)*i^2 + 10*((2*B^2*b^5*c^3*
d^2 - 15*B^2*a*b^4*c^2*d^3 + 60*B^2*a^2*b^3*c*d^4 - 47*B^2*a^3*b^2*d^5)*i^2*n + 60*(A*B*b^5*c^3*d^2 - 3*A*B*a*
b^4*c^2*d^3 + 3*A*B*a^2*b^3*c*d^4 - A*B*a^3*b^2*d^5)*i^2)*x^2 + 5*((27*B^2*b^5*c^4*d - 100*B^2*a*b^4*c^3*d^2 +
120*B^2*a^2*b^3*c^2*d^3 - 47*B^2*a^4*b*d^5)*i^2*n + 60*(3*A*B*b^5*c^4*d - 8*A*B*a*b^4*c^3*d^2 + 6*A*B*a^2*b^3
*c^2*d^3 - A*B*a^4*b*d^5)*i^2)*x + 60*(B^2*b^5*d^5*i^2*n*x^5 + 5*B^2*a*b^4*d^5*i^2*n*x^4 + 10*B^2*a^2*b^3*d^5*
i^2*n*x^3 + 10*(B^2*b^5*c^3*d^2 - 3*B^2*a*b^4*c^2*d^3 + 3*B^2*a^2*b^3*c*d^4)*i^2*n*x^2 + 5*(3*B^2*b^5*c^4*d -
8*B^2*a*b^4*c^3*d^2 + 6*B^2*a^2*b^3*c^2*d^3)*i^2*n*x + (6*B^2*b^5*c^5 - 15*B^2*a*b^4*c^4*d + 10*B^2*a^2*b^3*c^
3*d^2)*i^2*n)*log((b*x + a)/(d*x + c)))*log(e) + 60*((47*B^2*b^5*d^5*i^2*n^2 + 60*A*B*b^5*d^5*i^2*n)*x^5 + (72
*B^2*b^5*c^5 - 225*B^2*a*b^4*c^4*d + 200*B^2*a^2*b^3*c^3*d^2)*i^2*n^2 + 5*(60*A*B*a*b^4*d^5*i^2*n + (12*B^2*b^
5*c*d^4 + 35*B^2*a*b^4*d^5)*i^2*n^2)*x^4 + 60*(6*A*B*b^5*c^5 - 15*A*B*a*b^4*c^4*d + 10*A*B*a^2*b^3*c^3*d^2)*i^
2*n + 10*(60*A*B*a^2*b^3*d^5*i^2*n - (3*B^2*b^5*c^2*d^3 - 30*B^2*a*b^4*c*d^4 - 20*B^2*a^2*b^3*d^5)*i^2*n^2)*x^
3 + 10*((2*B^2*b^5*c^3*d^2 - 15*B^2*a*b^4*c^2*d^3 + 60*B^2*a^2*b^3*c*d^4)*i^2*n^2 + 60*(A*B*b^5*c^3*d^2 - 3*A*
B*a*b^4*c^2*d^3 + 3*A*B*a^2*b^3*c*d^4)*i^2*n)*x^2 + 5*((27*B^2*b^5*c^4*d - 100*B^2*a*b^4*c^3*d^2 + 120*B^2*a^2
*b^3*c^2*d^3)*i^2*n^2 + 60*(3*A*B*b^5*c^4*d - 8*A*B*a*b^4*c^3*d^2 + 6*A*B*a^2*b^3*c^2*d^3)*i^2*n)*x)*log((b*x
+ a)/(d*x + c)))/((b^11*c^3 - 3*a*b^10*c^2*d + 3*a^2*b^9*c*d^2 - a^3*b^8*d^3)*g^6*x^5 + 5*(a*b^10*c^3 - 3*a^2*
b^9*c^2*d + 3*a^3*b^8*c*d^2 - a^4*b^7*d^3)*g^6*x^4 + 10*(a^2*b^9*c^3 - 3*a^3*b^8*c^2*d + 3*a^4*b^7*c*d^2 - a^5
*b^6*d^3)*g^6*x^3 + 10*(a^3*b^8*c^3 - 3*a^4*b^7*c^2*d + 3*a^5*b^6*c*d^2 - a^6*b^5*d^3)*g^6*x^2 + 5*(a^4*b^7*c^
3 - 3*a^5*b^6*c^2*d + 3*a^6*b^5*c*d^2 - a^7*b^4*d^3)*g^6*x + (a^5*b^6*c^3 - 3*a^6*b^5*c^2*d + 3*a^7*b^4*c*d^2
- a^8*b^3*d^3)*g^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**6,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}^{2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x, algorithm="giac")

[Out]

integrate((d*i*x + c*i)^2*(B*log(e*((b*x + a)/(d*x + c))^n) + A)^2/(b*g*x + a*g)^6, x)